Wellposedness for the fourth order nonlinear Schrödinger equations

We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1Di∂tu=∂x4u+P((∂xαu)|α|⩽2,(∂xαu¯)|α|⩽2),t,x∈R, where P(⋅)P(⋅) is a polynomial excluding constant or linear terms.